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    <div class="post-body" itemprop="articleBody"><h1 id="模拟与高精度">模拟与高精度</h1>
<p>通过编程模拟生活中的活动、现象，是解决各种问题的基础。很多问题并不需要特定的算法去解决，往往有效利用各种数据类型，根据问题的背景建模为数据的表达，按部就班地操作，就可以得到答案。</p>
<p>超出 long long (<span class="math inline">\(2^64\)</span>)
范围的数字的加减乘除就是一类典型的模拟，称为高精度问题。</p>
<span id="more"></span>
<h2 id="基本的模拟">基本的模拟</h2>
<h3 id="例机器人的指令">例：机器人的指令</h3>
<p>数轴原点有一个机器人。该机器人将执行一系列指令，你的任务是预测所有指令执行完毕之后它的位置。</p>
<ul>
<li>LEFT：往左移动一个单位</li>
<li>RIGHT: 往右移动一个单位</li>
<li>SAME AS i: 和第i 条执行相同的动作。输入保证i
是一个正整数，且不超过之前执行指令数</li>
</ul>
<p>输入第一行为数据组数<span class="math inline">\(T (T \leq
100)\)</span>。每组数据第一行为整数<span class="math inline">\(n (1\leq
n\leq
100)\)</span>，即指令条数。以下每行一条指令。指令按照输入顺序编号为<code>1~n</code>。</p>
<p>对于每组数据，输出机器人的最终位置。每处理完一组数据，机器人应复位到数轴原点。</p>
<p>输入</p>
<figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><span class="line">2</span><br><span class="line">3</span><br><span class="line">LEFT</span><br><span class="line">RIGHT</span><br><span class="line">SAME AS 2</span><br><span class="line">5</span><br><span class="line">LEFT</span><br><span class="line">SAME AS 1</span><br><span class="line">SAME AS 2</span><br><span class="line">SAME AS 1</span><br><span class="line">SAME AS 4</span><br></pre></td></tr></table></figure>
<p>输出</p>
<figure class="highlight txt"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">1</span><br><span class="line">-5</span><br></pre></td></tr></table></figure>
<p>记录操作不一定要记单词，能区分它们就行，然后按照要求移动。</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;stdio.h&gt;</span></span></span><br><span class="line"><span class="type">char</span> op[<span class="number">111</span>], buf[<span class="number">111</span>];</span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>&#123;</span><br><span class="line">    <span class="type">int</span> t, n, x;</span><br><span class="line">    <span class="keyword">for</span>(<span class="built_in">scanf</span>(<span class="string">&quot;%d&quot;</span>, &amp;t); t --; ) &#123;</span><br><span class="line">        <span class="built_in">scanf</span>(<span class="string">&quot;%d&quot;</span>, &amp;n);</span><br><span class="line">        x = <span class="number">0</span>;  <span class="comment">// 多组测试数据，每次都要初始化</span></span><br><span class="line">        <span class="keyword">for</span>(<span class="type">int</span> i = <span class="number">0</span>; i &lt; n; i ++) &#123;</span><br><span class="line">            <span class="built_in">scanf</span>(<span class="string">&quot;%s&quot;</span>, buf);</span><br><span class="line">            <span class="keyword">if</span>(buf[<span class="number">0</span>] == <span class="string">&#x27;L&#x27;</span>) op[i] = <span class="string">&#x27;L&#x27;</span>;</span><br><span class="line">            <span class="keyword">else</span> <span class="keyword">if</span>(buf[<span class="number">0</span>] == <span class="string">&#x27;R&#x27;</span>) op[i] = <span class="string">&#x27;R&#x27;</span>;</span><br><span class="line">            <span class="keyword">else</span> &#123;</span><br><span class="line">                <span class="built_in">scanf</span>(<span class="string">&quot;%s&quot;</span>, buf);</span><br><span class="line">                <span class="type">int</span> ith;</span><br><span class="line">                <span class="built_in">scanf</span>(<span class="string">&quot;%d&quot;</span>, &amp;ith);</span><br><span class="line">                op[i] = op[ith - <span class="number">1</span>];</span><br><span class="line">            &#125;</span><br><span class="line">            x += op[i] == <span class="string">&#x27;L&#x27;</span> ? <span class="number">-1</span> : <span class="number">1</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="built_in">printf</span>(<span class="string">&quot;%d\n&quot;</span>, x);</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h2 id="高精度">高精度</h2>
<h3 id="大整数加法">大整数加法</h3>
<p>模拟小学学习的竖式，两个数组表示两个加数，另一个数组表示相加之和，数组每一项代表一位，下标<code>0</code>对应个位，<code>1</code>对应十位……</p>
<p>通常以字符串形式输入，然后处理为数字。</p>
<blockquote>
<p>知识点：字符的ASCII码（char存储）本身也是数字，只不过char是8位二进制，只能表示<code>-128~127</code>，只要在此范围内，把它当数字去计算是可以的。但要注意，<code>'1'</code>
的编码是<code>49</code>，要想让字符<code>'1'</code>成为数字<code>'1'</code>，可以由<code>'1' - '0'</code>
计算偏移量。</p>
</blockquote>
<p>以下代码仅供参考，具体实现可以有不同风格，选择自己易理解易记忆的方式。</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;stdio.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;string.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;algorithm&gt;</span></span></span><br><span class="line"><span class="type">char</span> a[<span class="number">511</span>], b[<span class="number">511</span>], c[<span class="number">511</span>];</span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>&#123;</span><br><span class="line">    <span class="built_in">scanf</span>(<span class="string">&quot;%s%s&quot;</span>, a, b);</span><br><span class="line">    <span class="type">int</span> alen = <span class="built_in">strlen</span>(a), blen = <span class="built_in">strlen</span>(b), clen = <span class="number">0</span>;</span><br><span class="line">    <span class="comment">// 读入的时候 [0] 存的高位，翻转过来，让[0]存个位，方便a和b对齐</span></span><br><span class="line">    std::<span class="built_in">reverse</span>(a, a + alen);</span><br><span class="line">    std::<span class="built_in">reverse</span>(b, b + blen);</span><br><span class="line">    <span class="keyword">for</span>(clen = <span class="number">0</span>; clen &lt; alen || clen &lt; blen; clen ++) &#123;</span><br><span class="line">        <span class="comment">// 字符ASCII码处理为数字</span></span><br><span class="line">        c[clen] = (clen &lt; alen ? a[clen] - <span class="string">&#x27;0&#x27;</span> : <span class="number">0</span>) + (clen &lt; blen ? b[clen] - <span class="string">&#x27;0&#x27;</span> : <span class="number">0</span>);</span><br><span class="line">        <span class="keyword">if</span>(clen &gt; <span class="number">0</span>) &#123;</span><br><span class="line">            c[clen] += c[clen - <span class="number">1</span>] / <span class="number">10</span>;</span><br><span class="line">            c[clen - <span class="number">1</span>] %= <span class="number">10</span>;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">if</span>(c[clen - <span class="number">1</span>] &gt; <span class="number">9</span>) &#123;</span><br><span class="line">        c[clen - <span class="number">1</span>] %= <span class="number">10</span>;</span><br><span class="line">        c[clen ++] ++;</span><br><span class="line">        </span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">for</span>(<span class="type">int</span> i = clen - <span class="number">1</span>; i &gt;= <span class="number">0</span>; i --) &#123;</span><br><span class="line">        <span class="built_in">printf</span>(<span class="string">&quot;%d&quot;</span>, c[i]); <span class="comment">// 这里c存的是“数字”`0,1,2`，而不是字符`&#x27;0&#x27;,&#x27;1&#x27;,&#x27;2&#x27;`，所以用%d输出</span></span><br><span class="line">    &#125;</span><br><span class="line">    <span class="built_in">printf</span>(<span class="string">&quot;\n&quot;</span>);</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="大整数乘法">大整数乘法</h3>
<p>比加法稍复杂一点，仍然是小学竖式的模拟。可以简化处理的是，可以把进位放在各位相乘之后再统一处理，每个位就只需要处理一次进位了。</p>
<p>以下代码仅供参考，具体实现可以有不同风格，选择自己易理解易记忆的方式。</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;stdio.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;string.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;algorithm&gt;</span></span></span><br><span class="line"><span class="type">char</span> a[<span class="number">2111</span>], b[<span class="number">2111</span>];</span><br><span class="line"><span class="comment">// 进位累加过程中可能 char 的128存不下，存结果用int</span></span><br><span class="line"><span class="comment">// 两数相乘结果的位数不小于两数位数之和，数组开大一倍</span></span><br><span class="line"><span class="type">int</span> c[<span class="number">4111</span>];</span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>&#123;</span><br><span class="line">    <span class="built_in">scanf</span>(<span class="string">&quot;%s%s&quot;</span>, a, b);</span><br><span class="line">    <span class="type">int</span> alen = <span class="built_in">strlen</span>(a), blen = <span class="built_in">strlen</span>(b), clen = <span class="number">0</span>;</span><br><span class="line">    std::<span class="built_in">reverse</span>(a, a + alen);</span><br><span class="line">    std::<span class="built_in">reverse</span>(b, b + blen);</span><br><span class="line">    <span class="built_in">memset</span>(c, <span class="number">0</span>, <span class="built_in">sizeof</span>(c));    <span class="comment">// 初始化 c 数组，确保全为 0</span></span><br><span class="line">    <span class="keyword">for</span>(<span class="type">int</span> i = <span class="number">0</span>; i &lt; alen; i ++) &#123;</span><br><span class="line">        <span class="keyword">for</span>(<span class="type">int</span> j = <span class="number">0</span>; j &lt; blen; j ++) &#123;</span><br><span class="line">            <span class="comment">// 比如 80 * 900，其 8（下标1） * 9（下标2） = 72 应该落在 72000（下标3）上</span></span><br><span class="line">            <span class="comment">// 进位可结束之后再统一处理</span></span><br><span class="line">            c[i + j] += (a[i] - <span class="string">&#x27;0&#x27;</span>) * (b[j] - <span class="string">&#x27;0&#x27;</span>);</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">for</span>(; clen &lt; alen + blen + <span class="number">5</span>; clen ++) &#123;</span><br><span class="line">        c[clen + <span class="number">1</span>] += c[clen] / <span class="number">10</span>;</span><br><span class="line">        c[clen] %= <span class="number">10</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="comment">// 去掉高位前导 0，且即使c是0也至少有 1 位数</span></span><br><span class="line">    <span class="keyword">for</span>(; clen &gt; <span class="number">1</span> &amp;&amp; c[clen - <span class="number">1</span>] == <span class="number">0</span>; clen --);    </span><br><span class="line">    <span class="keyword">for</span>(<span class="type">int</span> i = clen - <span class="number">1</span>; i &gt;= <span class="number">0</span>; i --) &#123;</span><br><span class="line">        <span class="built_in">printf</span>(<span class="string">&quot;%d&quot;</span>, c[i]);</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="built_in">printf</span>(<span class="string">&quot;\n&quot;</span>);</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

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